if the main character is the same (I cant remember) then yes, it is him going to Vice City to try and get the mafia a foothold in Vice City, the initial method doesnt work out and everything goes awol and then you gotta fix things.

GTA3 took place in Liberty City, with an unnamed character (he didn't talk too, to make the gamer feel he/she was really that character) and with no exact date set. But most people assume it took place in the late 90's/early 00's, the time the game was developed.
Vice City takes place in 1986, with a character named Tommy Vercetti. They now choose for a date, so that the player had the feeling to be in the middle of the top years of the drugs-syndicates, the late 80's. That also explains the music (all hits from that same time-period).
There is no real connection between the two games, the only connection is that Tommy happened to be in a prison in Liberty City (another interesting fact, he was released after 15 years of imprisonment, so he was brought to prison in 1971. GTA3 had modern cars, an internetcafe and modern music - that all didn't exist in or before 1971 - so it just can't be the same guy).